TY - JOUR

T1 - Generation of relative commutator subgroups in Chevalley groups

AU - Hazrat, R.

AU - Vavilov, N.

AU - Zhang, Z.

N1 - Publisher Copyright:
© Edinburgh Mathematical Society 2016.

PY - 2016/5/1

Y1 - 2016/5/1

N2 - Let Φ be a reduced irreducible root system of rank greater than or equal to 2, let R be a commutative ring and let I, J be two ideals of R. In the present paper we describe generators of the commutator groups of relative elementary subgroups [E(Φ,R,I),E(Φ,R,J)] both as normal subgroups of the elementary Chevalley group E(Φ,R), and as groups. Namely, let xα(ξ), α ϵ Φ ξ ϵ R, be an elementary generator of E(Φ,R). As a normal subgroup of the absolute elementary group E(Φ,R), the relative elementary subgroup is generated by xα(ξ), α ϵ Φ, ξ ϵ I. Classical results due to Stein, Tits and Vaserstein assert that as a group E(Φ,R,I) is generated by zα(ξ,η), where α ϵ Φ, ξ ϵ I, η ϵ R. In the present paper, we prove the following birelative analogues of these results. As a normal subgroup of E(Φ,R) the relative commutator subgroup [E(Φ,R,I),E(Φ,R,J)] is generated by the following three types of generators: (i) [xα(ξ),zα(ζ,η)], (ii) [xα(ξ),x-α(ζ)] and (iii) xα(ξζ), where α ϵ Φ, ξ ϵ I, ζ ϵ J, η ϵ R. As a group, the generators are essentially the same, only that type (iii) should be enlarged to (iv) zα(ξζ,η). For classical groups, these results, with many more computational proofs, were established in previous papers by the authors. There is already an amazing application of these results in the recent work of Stepanov on relative commutator width.

AB - Let Φ be a reduced irreducible root system of rank greater than or equal to 2, let R be a commutative ring and let I, J be two ideals of R. In the present paper we describe generators of the commutator groups of relative elementary subgroups [E(Φ,R,I),E(Φ,R,J)] both as normal subgroups of the elementary Chevalley group E(Φ,R), and as groups. Namely, let xα(ξ), α ϵ Φ ξ ϵ R, be an elementary generator of E(Φ,R). As a normal subgroup of the absolute elementary group E(Φ,R), the relative elementary subgroup is generated by xα(ξ), α ϵ Φ, ξ ϵ I. Classical results due to Stein, Tits and Vaserstein assert that as a group E(Φ,R,I) is generated by zα(ξ,η), where α ϵ Φ, ξ ϵ I, η ϵ R. In the present paper, we prove the following birelative analogues of these results. As a normal subgroup of E(Φ,R) the relative commutator subgroup [E(Φ,R,I),E(Φ,R,J)] is generated by the following three types of generators: (i) [xα(ξ),zα(ζ,η)], (ii) [xα(ξ),x-α(ζ)] and (iii) xα(ξζ), where α ϵ Φ, ξ ϵ I, ζ ϵ J, η ϵ R. As a group, the generators are essentially the same, only that type (iii) should be enlarged to (iv) zα(ξζ,η). For classical groups, these results, with many more computational proofs, were established in previous papers by the authors. There is already an amazing application of these results in the recent work of Stepanov on relative commutator width.

KW - Chevalley groups

KW - commutator subgroups

KW - elementary generators

UR - http://www.scopus.com/inward/record.url?scp=84945297715&partnerID=8YFLogxK

U2 - 10.1017/S0013091515000188

DO - 10.1017/S0013091515000188

M3 - Article

AN - SCOPUS:84945297715

SN - 0013-0915

VL - 59

SP - 393

EP - 410

JO - Proceedings of the Edinburgh Mathematical Society

JF - Proceedings of the Edinburgh Mathematical Society

IS - 2

ER -